You could do with providing some more details for this query.
For example, was your transfected material integrated into the cell genome or it is episomally expressed as a plasmid?
In the case of genomic integration you may be able to remove the selection drug after successful transfection, but it may still be useful to keep the drug in the media to prevent any contamination from non-transfected cells.
If you have transfected a plasmid, I would recommend keeping the drug in the media or else you may lose the plasmid. But, if you are only after transient expression you may be able to get away with removing the drug.
It also depends entirely on what your downstream applications are and whether the drug will interfere with those.
while undertaking stable transfection of cells it is advisable to culture such cells in Puromycin (or your selection marker) for 1-2 generations for stringent selection. After successful transfection (stable), you can withdraw your antibiotic from the medium as the untransfected cells have already been eliminated due to your drug treatment.
You can use the non-infected/-transfected cells as a control for puromycin selection. Once your non-infected/-transfected cells die off completely, then you will be sure that all selected cells are expressing the antibiotic-resistant genes. Sometimes, we use antibiotics for stable cells all the time in the media; however, for some stable cells, we do not use antibiotics in the culture media. We culture cells both ways.